Riesz sequence

In mathematics, a sequence of vectors (xn) in a Hilbert space is called a Riesz sequence if there exist constants such that

for all sequences of scalars (an) in the p space2. A Riesz sequence is called a Riesz basis if

.

Theorems

If H is a finite-dimensional space, then every basis of H is a Riesz basis.

Let be in the Lp space L2(R), let

and let denote the Fourier transform of . Define constants c and C with . Then the following are equivalent:

The first of the above conditions is the definition for () to form a Riesz basis for the space it spans.

See also

References

  • Christensen, Ole (2001), "Frames, Riesz bases, and Discrete Gabor/Wavelet expansions" (PDF), Bulletin (New Series) of the American Mathematical Society, 38 (3): 273–291
  • Mallat, Stéphane (2008), A Wavelet Tour of Signal Processing: The Sparse Way (PDF) (3rd ed.), pp. 46–47, ISBN 9780123743701

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